‘Hungarian Academy o f‘Sciences C E N T R A L R E S E A R C H IN S T IT U T E FOR P H Y S IC S B U D A P E S T

KFKI-1981-07

A, SÜTŐ

MAGNETIZATION IN SOME FRUSTRATION MODELS

_ _ _

‘H ungarian A cadem y o f ‘Sciences C E N T R A L

R E S E A R C H

I N S T I T U T E F O R P H Y S I C S

B U D A P E S T

A. Sütő*

Central Research Institute for Physics H-1525 Budapest 114, P.O.B. 49, Hungary

HU ISSN 0368 5330 ISBN 962 271 784 1

‘Present address: Universit4 de Lausanne Section de Physique CH-1015 Dorigny

magnetized equilibrium states, if the temperature is sufficiently low. Spins in the frustrated cells also carry a moment the magnitude of which is 0.528 at zero temperature.

АННОТАЦИЯ

В модели, предложенной Лонга и Олешом, существуют два равновесных состо­

яния с противоположно направленной намагниченностью, если температура доста­

точно низкая. Спины в фрустированной ячейке имеют магнитный момент, величина которого равна 0,528 при нулевой температуре.

KI VONAT

Longa és Oleá által javasolt modellekben két ellentétesen mágnesezett egyensúlyi állapot létezik, ha a hőmérséklet elegendően alacsony. A fruszt­

rált cellákban levő spinek szintén hordoznak mágneses momentumot, melynek értéke zéró hőmérsékleten 0.528.

1 . Introduction

Recently, Longa and Oles' (1980) studied a family of peri­

odic Ising frustration models on the square lattice, in which frustrated squares occupied pairs of neighbouring columns and two such pairs were separated by m H col­

umns of non-frustrated squares (fig.1). This distribution of frustration can be realized, for example, by choosing the bonds negative along each (m+2)th vertical line and positive otherwise. Applying the method of dimers they calculated the free energy of these models and found a singularity at some T=Tc (m) >0. To study the low temper­

ature behaviour, they performed a mean-field calcula­

tion which suggested the appearance of long-range order in areas of non-frustrated squares. In the present note, this suggestion is verified rigorously and it is shown that the spins in the frustrated cells also become par­

tially ordered.

Very recently, Hoever et al.(1980) extended the discus­

sion to models with arbitrary periodic distribution of columns of frustrated squares : they calculated the free energy and found a simple and striking condition for the existence of a positive critical temperature.

The study of the magnetization is more difficult in this case and will be the subject of future work.

On physical grounds, it is easy to understand why does magnetization set in at low temperatures in the models of Longa and Oles. Let us adopt the choice for the bonds as indicated above and consider the ground state spin configurations (g_s) of the system. The * 1 and

G" = -1 configurations are gs; on fig.1, full lines of unit length cross the negative bonds, indicating that

they are the "wrong bonds" in and s. : those at the higher energy level. Consider, e.g., C + . A local zero energy transformation (I z e t ) which consists of flipping several non-neighbouring spins along a vertical line with negative bonds, carries into another g_s. Let F + be the family of those g_s which can be obtained from

C + by performing a sequence of Izet and let F_ be the corresponding family for 6 _ . Every £s in F + (F_) shows long range order in the sense that every spin outside the negative vertical lines has the value +1

(-1). Clearly,-F =F+ and these sets are disjoint. One expects that F + and F_ are the continuations, to T=0, of oppositely magnetized low temperature phases. The complication arises from the existence of a family F^

of which is disjoint from both F + and F_ . The ele­

ments of F can be obtained from that of F , or F by flipping whole strips of spins; an example is shown on fig.2. The "strip-flip" transformation is not local but it can be performed as a sequence of local transforma­

tions at a total cost of energy proportional to the

width of the strip. Therefore, Fq provides with a channel between F + and F_ , available at whatever small positive temperatures. A simple numerical estimate shows, however, that the mixing of F + and F_ via Fq is a negligible

effect: It is easy to calculate the total number of gs. * IF tl , and the number of срз with long range order,

IF+l + IF i . Considering a square of N sites, one finds the asymptotic results

and

tot = |F + M F _ M F qI =2 (1+cVRj 'Ш/ (m+2)

|F + | + IF I = 2 cN/<m+2)

where c=(1 + 'T5)/2 (see later in the text). Hence, the entropy of the mixing at zero temperature is

ln,Ftot' - ln(IP+ l + IP_l)= Ш М * С - Г"|

which vanishes in the thermodynamic limit, suggesting that F + and F_ represent different low temperature p h a s e s .

In Section 2, we make this "physical argument" precise.

In order to obtain this goal, we extend the method of Peierls which cannot be applied to the present problem neither in its original form (Peierls 1936), nor in a recent generalized version aimed to cover cases of frus­

tration (Sütő 1980).

2. Study of the magnetization

We consider any model with frustrated squares distributed as discussed above, for some m i l . Let be one of the

о

two 2J3 in which the wrong bonds are those along the ver­

tical lines between neighbouring columns of frustrated squares (see fig.1). We prove the following proposition:

If the temperature is sufficiently low then there exists an equilibrium state, belonging to in the following s e n s e :

(i) In any typical configuration, S’ , of this state, one can find an infinite connected set of sites over which

<r .

--- O —

(ii) If x is not a common site of four frustrated squares then

C (x) <€>(x)> „ > 0

о о

and goes to 1 with T going to 0.

(iii) If x is the common site of four frustrated squares (i.e./ x is in a frustrated cell) then

<еГ(х)> = [1-2(2/(1 + f5) )* 1 2 3] 6^ (x) »0.528 <5?

(

)

^ о at T=0.

We may remark the followings:

1) The equilibrium state belonging to 6Г can be gener­

ated as the thermodynamic limit of probability distribu­

tions in finite volumes, if 6(x)= 6" (x) on the boun- o

dary of these volumes. The notation > refers to this construction.

2) Except (iii), the above proposition contains the usu­

al statements which can be obtained by a Peierls type argument. A bound Tq , below which (i) and (ii) are veri­

fied and which is common for any m > 1, can be inferred from the proof; this Tq is, however, a poor lower esti­

mate for the critical temperatures.

3) By reason of symmetry, there exists another equilib­

rium state belonging to - 6* in the above sense. There­

fore, the properties (i)-(iii) imply the breakdown of the S'—* -6* symmetry of the Hamiltonian.

To prove the proposition, we consider a finite part V of the lattice, fix the configuration 6"^ outside V and study the equilibrium probability distribution Py for the configurations inside. By definition,

Pv [the configuration is ® in V]

<3 о

-1

exp (- p(H( « )-H( S

) ) ]

= Z -1

V, G> exp [~2|i 21 Jx v 5 o ( x ) & o (y)1 6су>*Э«?) ХУ ° °

(1)

where Ъ ( G1 ) contains those bonds <xy> for which

в (x) О (у) = - ^ o (x) в"о (y) and Zy G is the partition func­

tion corresponding to the boundary condition. If for each <хуНЭ(бг) one draws a dashed line of unit length crossing the bond <xy>, one finds that 'З(в) is represen­

ted by a collection of closed lines separating the d o ­ mains of V where б = 6Г from those where S'= - СГ ( a n

о о

example for ©(S') is shown on fig.3). Once is fixed outside V, there is a one-to-one correspondence between the configurations and the collections of closed lines on the dual lattice. Let _Q-o denote the set of the wrong bonds of ^ ; these are crossed by full lines of the

о

dual lattice. Now, the sum in the exponent of Eq.(1) has a simple geometric interpretation. The sets *Э( & ) and _flQ may have common bonds which appear on the figure as

coinciding full and dashed lines; if I ’ S (1 JlQ l is the number of common bonds and l© - _fL I is the number of

о

bonds belonging to © but not to J1q , then

V V * > = -1'э п -л о и к -э

^ху>«Э ₍₂₎

Here we assumed that IJ 1=1. Plainly,

0 *■ Í |Э| - (3)

We say that a set of bonds,

Г

Г» Э( s )

6

Г

or

tiply connected line. For any <o , Ъ ( ef ) is the union

of maximal connected parts each of them being a contour.

If fore some x « V we find €(х)=-в' (x) then there is о

at least one contour in Э ( er ) which surrounds x. In the following formulas, Г always denotes a contour and

x € Int Ъ means that some part of Э surrounds x. Let now 0 t < 1 and x be an arbitrary site in V. Then the fol­

lowing inequalities are true for the probability distri­

bution (1) .

Pv t 6 (x) =- er o (x) ] é Py [x e Int Ъ ( & ) ]

á

Г :x (r Int Г

V

р [ Г ] + Z 1 р г г ] + 2 Z р t Г ] Г : х e Int Р Г : х е IntP Р : х € IntP

^ р i t SГI О < k p < €.|Г1 к р =0

= Ах ( £ , > ) A J С , О А х (0>

(4)

where Pv [ Р ] is the probability that Г is a maximal c o n ­ nected component of some Ъ ( G ). According to the usual Peierls argument (see, e.g., Griffiths 1972),

A ( Í , > ) ^ Z I nf/2) 3« e _2/5E€ (5) x € > 4

The second and third sums in Eq.(4) do not appear if

there is no frustration present; below we elaborate their es t i m a t e s .

Л

Let Э denote the set of those sites of the dual lattice which are visited by Ъ and let us introduce the notation

z=e -2[i1 . Then

Pv t Г ]=z

I . л л

э

» * / Z ^{(6 )}

where the denominator is just Z„ _ and, if some config- V , ®o

uration 6 contributes to the l.h.s., then Ъ (6 )= Э ' ^ Г with one of the Э* in the numerator of the r.h.s..

It is obviously true that

k Э* Э'

_ z > 2L. z z

•* Э': “Э 1 Э сГ

which gives us

к k ~

P [ P ] Í z / 2 . z

V _ Л Л

г

This estimate is valid for any contour Г

A„ ( l , < ) £

P :x € IntP 0 < к p <tlPI

( X 1) 8:§C P к э =°

- 1

At first, we show that

„ г ^ П - з е И П -1

= f ( iri )

Then

Э • Э с P к 3 = 0

holds for any P satisfying the inequalities

(7)

(8)

(9)

0 < k p < £ IT! (1 0)

For, let Г be such a contour. We consider the line rep­

resenting P (fig.3) and divide it into zero energy seg­

ments (zes) and purely positive energy segments (p p e s ) .

A zes is a maximal piece of Г which begins with a wrong bond, goes on with an alternating sequence of good and wrong bonds and is terminated by a good bond ( good and wrong bonds are elements of Г - Si and Г

П SL

o о

p e c t i v e l y ) . A ppes is a maximal connected part of Г b e ­ tween two z e s , therefore it contains only good bonds. The following elementary relations hold:

Г р = [number of zes] = [number of p p e s ] £ к г £е.|П|

(1 - C) 1 Г 1 /2 < i Г Л Л _ I 1Г1/2 (11) о

and, as a consequence,

Iz e s I=2 ■ [number of wrong bonds zes 6 P , I zesi * 4

belonging to zes of length >4]

* 2 ( IP a S L o l - r p ) > (1-3 £ ) IPI (12)

where Izesidenotes the length of z e s .

Now consider a zes of length 2t where C>2; this goes through the centers of 2( frustrated squares. These centers surround €-1 sites, x^,...,x^_^ , of the lattice

(denoted by circles on fig.3); the spins sitting here are in frustrated cells. To obtain the estimate (9) we have to calculate n ^ ^ , the number of ground states of these {-1 spins with the condition that the configura­

tion is outside them. Clearly, { er (x^) , . . . , 6"0 (x^_^)}

is a c[s and any configuration, not containing the detail . . . , - S (x^),- (x ) , . . . is also a ^ s . It is easy to see that n £ satisfies the difference equation for the Fibonacci numbers:

I

n « + 1 - 2 n t_, *(nt - n < _ 1)=n{_ 1 + n t (13a) with the initial conditions

n, = 2, n 2 = 3 (13b)

This equation can be solved by the use of the method of generating functions, resulting in

n

i

l + S + * ( Js + n *

S+fS { i I

И )

5 -is > № ) ⁽¹⁴⁾

for any { ± 1. Any zes the length of which is 2t > 4, contributes to rip with a factor n^_^ . From (12) and (14) then one obtains (9). The bound given in (9) depends only on the length of

P

nue (8) as

A ( t , < ) í Z. N, / f m (15)

X l 1

where denotes the number of contours of length i which surround x and satisfy (10). Now we give an upper bound to this number. It is easy to estimate the number of those contours which contribute to N ^ and contain a given bond, b, a given number of wrong bonds, and a given number of zero energy segments, r. Their number will be denoted by N^(b, l ,r ) . Starting from b, one can order the £. bonds of the contour in a sequence so that neighbouring bonds are joining in a site of the dual lattice. Therefore contours correspond to random walks of length i , starting from b. In each site along a p p e s , there is at most three possibilities to continue the walk; once the walk arrives at a zes, there is al-

together 6 possibilities until we can continue with the following ppes : 2 ways to choose the first good bond of the zes and 3 to choose the last one. The total length of the purely positive energy segments is ~Z (o , there­

fore

N,(b, Í , г Ш ' 6^ £ 3' ' ‘ *'u 6

= 18

a

(16)

where we used (11). It follows also from (11) that there are at most £ t and l i /2 different possibilities for choosing r and , respectively. Furthermore, if one starts from x and makes 1 / 2 steps to right, one certainly crosses at least one bond of any contour contributing to

. Whence, it is sufficient to choose the starting bond b from a set containing t / 2 bonds. These facts and (16) yield

£ j { 3 Í 2 1 8lt (17a)

Also, (11) gives

N = 0 if I c Л /b (17b)

because Гр > 1 for any Г satisfying (10). Equations (8), (9), (15) and (17) together result in

2 __ t ^

A ( £ , < ) < 0.4 C (37 ■ 0.787)

X « V l /t (18)

Suppose now that x is not in a frustrated cell; then A ^ (0)=0 and

P [€T(x) =-er (х) ]

í 2.

V ° l > 4

l /2 • 3*- -2 AE í e 1

+ 0.4 t (37 - 0.787)

<>1/E

(19)

Choosing £. = 0.066 and fi > 3 . 2 we find that the sums on the r.h.s. of (19) are convergent. Then, from the Borel- Cantelli lemma (see,e.g., Feller 1968) it follows that with probability 1 there is only a finite number of con­

tours surrounding x, which is another way to formulate the percolation property (i) of the proposition. If £ is so small that A^( ( , c ) < 1 /2- * (where «■ > 0) and (i is so large that A ( £ , > ) <£ <x/2 then

X

Py [6(x) =-erQ (x) ] é (I-« ) /2

showing that a moment, parallel to Sq (x), appears in x, Finally, if we keep £ fixed and let fb go to infinity, we obtain that

limsup P [6(x)=-er (x) ] é 0.4 t 51

(i-ое» ° £?1/i

i l

(37 - 0.787) (20)

This inequality is true for whatever positive t and volume V, implying that 6 (x) = 6'0 (x) at T=0 with full probability. This concludes the proof of the statement

(ii) of the proposition.

The bounds (5) and (18) are also valid for x not being in a frustrated cell. However, A (0) is not zero in that case. Let J(x) denote the shortest possible contour

around x, that is, the contour of the four edges sepa­

rating x from its nearest neighbours. Now, an<^

kp > 0 for any other contoiir around x. We can write

therefore

A x (0) = Pv [ tf(x)]

Below we show that

lim Ру [ tf(x)]= « 0.236 V-*oo

at T=0. Indeed, for /Ъ=+<х> ,

(2 1⁾

(2 2⁾

if к Э > °

and the substitution of (23) into (6) gives

(23)

Pv [jf(x)]= 1/ 1 (24)

Ъ :У (x)с Э ?:k =0

In the numerator , the summation runs over those £s which coincide with $ outside V and - €T on the site x. In

о о

the denominator, we. find the same summation except the restriction on S'(x). In every g_s occurring in these sum­

mations the configuration outside the frustrated cells coincides with 6 . The number of gs is therefore the

о -2 —

product of the numbers of £s in each column of frustra­

ted cells. The contribution of every column cancels out in (24), except that of the one containing x. If, in this column, there are m^ site above x and m 2 below it, then

P„[X(x)]= n n - / n_ ._ . - V u m^-1 m 2~' m i+ m 2+ '

where n ^ is given by (14). If both m^ and m2 goes to infinity, we obtain the limit (22). The third part of the proposition follows from (22) and the fact that

(25)

lim Ру [ S’(x) =-6T Q (x) ] = lim Ру [у(х)] (26)

(i -»oo ft-’ oo

3. Concluding remarks

We have rigorously shown that frustrated systems descri­

bed by the above models become magnetically ordered at sufficiently low temperatures. An interesting finding is that ground states which locally transform into each other may not be equivalent from a statistical point of view. Spins in the frustrated cells become magnetized though their moments are not fully saturated at zero temperature. Therefore, a periodic oscillation of the magnetization appears in the horizontal direction. The continuity, at T=0, of the moments in the frustrated cells still needs a proof.

References

Feller W 1968 An Introduction to Probability Theory and Its Applications (Wiley,New York) Vol.1.

Griffiths R В 1972 in Phase Transitions and Critical Phenomena, C.Domb and M.S.Green, eds.

(Academic Press, New York) Vol.1.

Hoever P, Wolff W F and Zittartz J 1980 "Random Layered Frustration Models" Preprint, to appear in Z.Physik В

Longa L and Oleá A M 1 980 J . Phys . A :M a t h .Gen ._L3 1031-42 Peierls R 1936 Proc .Camb. P h i l . Soc . 32^ 477

SÜtó A 1980 J.Stat.Phys. 23 203-17

Fig. 1

The m=l frustration model. Crosses mark frustrated squares and the lines connecting them indicate the

wrong bonds of the ground states o+ and o _ (a , in general).

Fig. 2

Wrong bonds in a ground state belonging to F .

1 I f ■

— - <

l- .

* -f

1 /, 1 1

- J L .

~ \ t

Л

, j

9 V

L_ .

■ 4 -

_ ( L . _ J

Fig. 3

The ground state aQ and a contour with respect to it.

Zero energy sequences are put in parentheses.

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Kiadja a Központi Fizikai Kutató Intézet Felelős kiadó: Krén Emil

Szakmai lektor: D r . Siklós Tivadar Nyelvi lektor: Dr. Sólyom Jenő Példányszám: 520 Törzsszám: 81-62 Készült a KFKI sokszorosító üzemében Felelős vezető: Nagy Károly

Budapest, 1981. február hó

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